Isn't it more reasonable to assume they do not exist? Further, that a neutron is made of a proton and electron rather than 3 quarks? How could we determine this (such as firing neutrons at a strong magnetic field and seeing if half half split due to difference in electron/proton orientation)?
The fact that pointlike subcomponents of hadrons exist with the quantum numbers of quarks is experimentally verified in what is called the parton model.
Basically, you assume hadrons are bags of pointlike "partons" held together by a very strong force which becomes weak at very high energy. When you have a high-energy collision between hadrons (example: proton-proton collision) what happens is the partons are essentially free. You could also say the interaction is so fast they have no time to realize they are part of a hadron. So only two partons, each from one of the hadrons, actually interact. Each parton carries a fraction (a part) of the whole energy and momentum of the hadron; in general you have a certain probability to find a parton carrying a fraction 0 < x < 1 of the whole momentum of the hadron.
So the whole probability of interaction will be the probability that two free partons would interact weighted with the probability of finding that kind of parton with that energy in your hadrons.
This model has (had really, this is very old stuff) excellent agreement with experimental data in proton-proton collisions. You can extract the quantum numbers of the partons, like the charge or the spin. So hadrons are literally made of partons and you call them quarks.
You can also deduce the existence of the colour force; only a force of this kind has the particular behaviour of becoming weak at high energy, which is asymptotic freedom.
A more recent and almost direct proof of quarks and gluons is given by hadronic jets. Some collisions are so energetic that they produce almost free real gluon or quarks as final products. As soon as the products get separated enough though the colour force works towards "dressing" the gluon and quarks to make them colourless (coloured particles are not allowed by the strong force!) and so they "hadronize" and they are turned into showers of multiple hadrons. However we can measure the single jets of hadrons and reconstruct the original momenta of the gluons and quarks.Source